sec:prelim summarises definitions and notations used in this paper.
In sec:obstructions we define the classes \(\mathcal {F}_=,\mathcal {F}_\leqslant ,\mathcal {F}_\geqslant \)  of
obstructions to twin-width in tournaments, and prove thm:forbidden-tournaments-bad.
sec:bst defines binary search trees, the associated orders, and some related notions.
We then prove a crucial lemma which from a partition into intervals of a BST order
extracts some FO definable substructure.
sec:extraction then uses this lemma, combined with results of [1]} and Ramsey arguments,
to prove our main technical result, thm:forbidden-tournaments-intro.
Finally, sec:generalisation explains how our results generalise
to oriented graphs with bounded independence number, and to binary relational structures.
